Analysis of Boston data
Jussi Jaatinen
Analysis of Boston data
Overview of data
Boston data is included in R-package as a demonstration or example.
Dataset contains social, environmental and economical information about great Boston area. It includes following variables:
- crim = per capita crime rate by town
- zn = proportion of residential land zoned for lots over 25,000 sq.ft.
- indus = proportion of non-retail business acres per town
- chas = Charles River dummy variable
- nox = nitrogen oxides concentration (parts per 10 million)
- rm = average number of rooms per dwelling
- age = proportion of owner-occupied units built prior to 1940
- dis = weighted mean of distances to five Boston employment centres
- rad = index of accessibility to radial highways
- tax = full-value property-tax rate per $10000
- ptratio = pupil-teacher ratio by town
- black = 1000(Bk - 0.63)^2 where Bk is the proportion of blacks by town
- lstat = lower status of the population (percent)
- medv = median value of owner-occupied homes in $1000s
Structure and the dimensions of the data
Dataset has 14 variables and 506 observations and all variables are numerical.
'data.frame': 506 obs. of 14 variables:
$ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
$ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
$ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
$ chas : int 0 0 0 0 0 0 0 0 0 0 ...
$ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
$ rm : num 6.58 6.42 7.18 7 7.15 ...
$ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
$ dis : num 4.09 4.97 4.97 6.06 6.06 ...
$ rad : int 1 2 2 3 3 3 5 5 5 5 ...
$ tax : num 296 242 242 222 222 222 311 311 311 311 ...
$ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
$ black : num 397 397 393 395 397 ...
$ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
$ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...[1] 506 14Summary, graphical presentation of data and correlations
As seen in pairs plot, most of the variables are not normally distributed. Most of them are skewed and some of them are bimodal. Correlations between variables are better viewed in correlation plotting, where on the upper-right side the biggest circles indicate highest correlations (blue = positive or red = negative). Corresponding number values are mirrored on the lower-left side.
crim zn indus chas
Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
1st Qu.: 0.08204 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
nox rm age dis
Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
rad tax ptratio black
Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
Median : 5.000 Median :330.0 Median :19.05 Median :391.44
Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
lstat medv
Min. : 1.73 Min. : 5.00
1st Qu.: 6.95 1st Qu.:17.02
Median :11.36 Median :21.20
Mean :12.65 Mean :22.53
3rd Qu.:16.95 3rd Qu.:25.00
Max. :37.97 Max. :50.00 
Standardization
In standardization means of all variables are in zero. That is, variables have distributed around zero. This can be seen in summary table (compare with original summary above).
Variable crime rate has been changed to categorical variable with 4 levels: low, med_low, med_high and high. Each class includes quantile of data (25%).
Train and test sets have been created by dividing original (standardized) data to two groups randomly. 80% belongs to train set and 20% to test set.
crim zn indus
Min. :-0.419367 Min. :-0.48724 Min. :-1.5563
1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668
Median :-0.390280 Median :-0.48724 Median :-0.2109
Mean : 0.000000 Mean : 0.00000 Mean : 0.0000
3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150
Max. : 9.924110 Max. : 3.80047 Max. : 2.4202
chas nox rm age
Min. :-0.2723 Min. :-1.4644 Min. :-3.8764 Min. :-2.3331
1st Qu.:-0.2723 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366
Median :-0.2723 Median :-0.1441 Median :-0.1084 Median : 0.3171
Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
3rd Qu.:-0.2723 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059
Max. : 3.6648 Max. : 2.7296 Max. : 3.5515 Max. : 1.1164
dis rad tax ptratio
Min. :-1.2658 Min. :-0.9819 Min. :-1.3127 Min. :-2.7047
1st Qu.:-0.8049 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876
Median :-0.2790 Median :-0.5225 Median :-0.4642 Median : 0.2746
Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
3rd Qu.: 0.6617 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058
Max. : 3.9566 Max. : 1.6596 Max. : 1.7964 Max. : 1.6372
black lstat medv
Min. :-3.9033 Min. :-1.5296 Min. :-1.9063
1st Qu.: 0.2049 1st Qu.:-0.7986 1st Qu.:-0.5989
Median : 0.3808 Median :-0.1811 Median :-0.1449
Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
3rd Qu.: 0.4332 3rd Qu.: 0.6024 3rd Qu.: 0.2683
Max. : 0.4406 Max. : 3.5453 Max. : 2.9865 Linear discriminant analysis (LDA)
In linear discriminant analysis (LDA) only the train set (80% of data) has been analysed. Target variable is the new categorical variable, crime rate (low, med_low, med_high, high). In LDA model all other variables of the data set are used as predictor variables (see Overview of data).
In biplot below can be seen that variable “rad” (index of accessibility to radial highways) has extremely high influence to LD1 and LD2 if compared to the other variables. In biplot all horizontal vectors describes contribution to LD1 dimension (x-axis) and vertical vectors LD2-dimension (y-axis). Sign of coefficient of linear discriminant determines the direction of vector. The longer the vector, the bigger is influence. Most of the vectors contribute both LD1 and LD2. Because in biplot two dimensions are illustrated, directions of most of variables are in different angles between LD1 and LD 2. For example, in the LDA table below the most significant variable of LD1 “rad” has coefficients LD1 = 3.27 and LD2 = 1.05. They are directly readable as coordinates of the arrow head. Similarly the second most significant variable of LD2, “nox” has its head ccordinates in (-0.69, 0.29). LDA1 alone explains 0.95% of model. LD2 explains 3% and LD3 only 1%.
Call:
lda(crime ~ ., data = train)
Prior probabilities of groups:
low med_low med_high high
0.2549505 0.2475248 0.2425743 0.2549505
Group means:
zn indus chas nox rm
low 0.92104479 -0.9126143 -0.081207697 -0.8836562 0.4511518
med_low -0.07454775 -0.2618785 0.003267949 -0.5995638 -0.1440819
med_high -0.38923530 0.1101206 0.209764839 0.3009757 0.2056717
high -0.48724019 1.0170891 -0.081207697 1.0513605 -0.3686680
age dis rad tax ptratio
low -0.8674806 0.8740330 -0.6986570 -0.7404913 -0.45706193
med_low -0.3748251 0.4351617 -0.5534930 -0.4961350 -0.05983227
med_high 0.3894962 -0.3501508 -0.4111534 -0.3380374 -0.23068070
high 0.8070771 -0.8623228 1.6384176 1.5142626 0.78111358
black lstat medv
low 0.37969838 -0.77606084 0.53976567
med_low 0.36047995 -0.12968145 -0.01357018
med_high 0.09979493 -0.07031796 0.20324164
high -0.75734598 0.89208473 -0.71601163
Coefficients of linear discriminants:
LD1 LD2 LD3
zn 0.07792721 0.64288658 -0.902748272
indus 0.04959931 -0.20772648 0.617025867
chas -0.11465325 -0.04849382 0.072183179
nox 0.40281708 -0.87907748 -1.267917206
rm -0.12150992 -0.13807916 -0.176351216
age 0.19993306 -0.33397579 -0.174569825
dis -0.04670761 -0.24812892 0.395086209
rad 3.31750902 0.90879821 -0.005502969
tax 0.08021468 0.08036632 0.335706010
ptratio 0.11111058 -0.02898242 -0.257307110
black -0.12728145 0.04958585 0.163897982
lstat 0.24809721 -0.14089168 0.598299636
medv 0.21743004 -0.35199369 -0.003050695
Proportion of trace:
LD1 LD2 LD3
0.9573 0.0299 0.0128 
Predictive power of the model
In the test dataset catecorigal crime variable has been removed. In the table below true values of the original test data and predicted values of the test data (crime removed) are cross-tabulated. Total amount of observations is 102 (506/5 +1). In the table on diagonal axis (from top-left corner) are true values (sum = 76) and all others are predicted values (sum = 26). Prediction error is 26/102 ≈ 0.25
predicted
correct low med_low med_high high Sum
low 17 7 0 0 24
med_low 3 14 9 0 26
med_high 1 3 22 2 28
high 0 0 0 24 24
Sum 21 24 31 26 102Calculation of distances between the observations and optimal number of clusters
In this model euclidean distance matrix has been calculated. Results can be seen in table below. By using K-means algorithm, the optimal number of clusters can be investigated. When TWSS (total within sum of squares) drops significally, it indicates optimal number of clusters. In this case optimal number of clusters is 2 or 3. In the first plotting, data has classified into two and in the second plotting three clusters.
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.1343 3.4625 4.8241 4.9111 6.1863 14.3970 
Bonus
Here LDA is calculated with the clusters as target classes. All other variables in the Boston data are predictor variables. In LDA tables and biplots, differences between number of clusters can be seen. Depending on number of clusters, meaningful variables are different, as seen in plottings.
Call:
lda(clu3 ~ ., data = boston_scaled_new)
Prior probabilities of groups:
1 2 3
0.3083004 0.4130435 0.2786561
Group means:
crim zn indus nox rm age
1 0.8647905 -0.4872402 1.0949412 1.1524404 -0.4717159 0.78718751
2 -0.3763256 -0.3947158 -0.1595373 -0.2992872 -0.2772450 0.02424663
3 -0.3989734 1.1241495 -0.9749470 -0.8314162 0.9328502 -0.90687091
dis rad tax ptratio black lstat
1 -0.8532750 1.3172714 1.3530841 0.57186480 -0.6936034 0.88830984
2 0.0304962 -0.5856776 -0.5482746 0.09789185 0.2769803 -0.03824517
3 0.8988453 -0.5892747 -0.6843385 -0.77780359 0.3568316 -0.92612124
medv
1 -0.6893319
2 -0.1525634
3 0.9888052
Coefficients of linear discriminants:
LD1 LD2
crim 0.03403154 0.1907082
zn 0.04097709 0.8724398
indus -0.41998138 -0.1847419
nox -1.06730082 0.6654667
rm 0.26776386 0.5104809
age 0.21123453 -0.4670361
dis -0.07658281 0.3338914
rad -1.17679746 0.3612912
tax -0.96644205 0.5042929
ptratio -0.06864005 -0.1601563
black 0.07085932 -0.0359700
lstat -0.26379244 0.3189355
medv 0.01076674 0.6395889
Proportion of trace:
LD1 LD2
0.8729 0.1271 Call:
lda(clu4 ~ ., data = boston_scaled_new)
Prior probabilities of groups:
1 2 3 4
0.4011858 0.1660079 0.3221344 0.1106719
Group means:
crim zn indus nox rm age
1 -0.3793592 -0.3541732 -0.2369409 -0.3788647 -0.3076659 -0.09191194
2 -0.4124621 1.9031602 -1.0764395 -1.1428600 0.6095020 -1.39546865
3 0.8082769 -0.4872402 1.1165562 1.1413403 -0.4676591 0.79696608
4 -0.3587926 -0.1526454 -0.7764063 -0.2344411 1.5622579 0.10664320
dis rad tax ptratio black lstat
1 0.1379588 -0.5920713 -0.5838167 0.08129234 0.2784401 -0.06159606
2 1.5205894 -0.6250261 -0.5943244 -0.67561813 0.3537454 -0.90687350
3 -0.8539968 1.2199444 1.2927317 0.58616084 -0.6486732 0.87910380
4 -0.2952439 -0.4671119 -0.7549503 -0.98740427 0.3481389 -0.97522401
medv
1 -0.1690186
2 0.6658984
3 -0.7034406
4 1.6613592
Coefficients of linear discriminants:
LD1 LD2 LD3
crim -0.03692074 -0.063843487 0.17437690
zn -0.10469860 -1.680125461 -0.02034829
indus 0.62305615 -0.375784264 -0.51749840
nox 1.07818726 -0.501408664 0.49444666
rm -0.13015732 -0.062534323 0.64219861
age -0.18778510 0.593603356 0.12209172
dis 0.01495296 -0.528845086 -0.16022047
rad 0.71521529 0.091708443 0.26711660
tax 0.86706461 -0.832332218 0.24904935
ptratio 0.21355380 -0.110643988 -0.18716017
black -0.01780187 0.008088268 -0.03064577
lstat 0.23409949 -0.104352523 0.29032605
medv -0.14661672 0.063424182 0.97435297
Proportion of trace:
LD1 LD2 LD3
0.7143 0.2084 0.0773 Call:
lda(clu5 ~ ., data = boston_scaled_new)
Prior probabilities of groups:
1 2 3 4 5
0.10671937 0.06916996 0.39328063 0.23913043 0.19169960
Group means:
crim zn indus nox rm age
1 -0.2753323 -0.4872402 1.5337294 1.1273809 -0.6003284 0.9334996
2 1.4802645 -0.4872402 1.0149946 0.9676887 -0.2969389 0.7656016
3 -0.3884901 -0.3308141 -0.4873088 -0.4761310 -0.2318056 -0.1989968
4 -0.3981339 1.2930469 -0.9902994 -0.8283387 1.0566896 -0.9121115
5 0.9128084 -0.4872402 1.0149946 1.0333132 -0.4012324 0.7501117
dis rad tax ptratio black lstat
1 -0.8995039 -0.6096828 0.01485481 -0.3917541 -0.1262348 0.6474932
2 -0.8580043 1.6596029 1.52941294 0.8057784 -3.2970564 1.1699052
3 0.2356027 -0.5732709 -0.60914944 0.1061891 0.3164212 -0.1478389
4 0.9121798 -0.5955687 -0.67325561 -0.8788401 0.3554817 -0.9544043
5 -0.8108798 1.6596029 1.52941294 0.8057784 0.1673459 0.7112530
medv
1 -0.4618433
2 -1.0473739
3 -0.1012054
4 1.0995470
5 -0.5289453
Coefficients of linear discriminants:
LD1 LD2 LD3 LD4
crim 0.11743258 0.04770450 -0.14300153 -0.16024852
zn 0.34092540 0.12344732 -0.08107374 -1.23725268
indus 0.53753143 -1.85979858 0.92454526 -1.10944073
nox -0.00906438 -0.29233935 0.67637180 -0.69383481
rm -0.05331958 0.27825890 -0.27651832 -0.52558718
age -0.02982587 -0.24810645 0.11616057 0.17202368
dis -0.18913313 -0.03732336 0.16315838 -0.26855975
rad 5.88697673 2.04722673 0.02514581 0.50932714
tax 0.13814687 0.14410388 0.09950265 -0.25979580
ptratio 0.20734779 0.06893525 -0.07716785 0.22476182
black -0.33376654 1.18823488 2.14340012 -0.03614042
lstat -0.01997449 0.13151481 -0.19758426 -0.25835224
medv -0.11300541 0.24641561 -0.16457165 -0.62281824
Proportion of trace:
LD1 LD2 LD3 LD4
0.8192 0.0840 0.0681 0.0287 Call:
lda(clu6 ~ ., data = boston_scaled_new)
Prior probabilities of groups:
1 2 3 4 5 6
0.30237154 0.12845850 0.21146245 0.09683794 0.05731225 0.20355731
Group means:
crim zn indus nox rm age
1 -0.3975627 -0.1637000 -0.5873353 -0.6615915 -0.1682976 -0.6574663
2 -0.4140702 2.2813035 -1.1562550 -1.1768217 0.7293996 -1.4086475
3 -0.3231486 -0.4822312 0.6308150 0.5041272 -0.5223445 0.7753457
4 -0.3680230 -0.1494734 -0.7440331 -0.2107180 1.7049351 0.1966439
5 3.0022987 -0.4872402 1.0149946 1.0593345 -1.3064650 0.9805356
6 0.5173303 -0.4872402 1.0149946 1.0036872 -0.1109215 0.6904986
dis rad tax ptratio black lstat
1 0.5448232 -0.5720261 -0.6911565 -0.06248294 0.36026575 -0.3806108
2 1.5645584 -0.6656010 -0.5702843 -0.80946918 0.35416061 -0.9741045
3 -0.5711540 -0.5943977 -0.1915536 0.08745080 0.03673407 0.5492158
4 -0.3113322 -0.5037339 -0.7871603 -1.09274698 0.34883211 -0.9623279
5 -1.0484716 1.6596029 1.5294129 0.80577843 -1.19066142 1.8708759
6 -0.7599982 1.6596029 1.5294129 0.80577843 -0.62752658 0.5406100
medv
1 0.01419929
2 0.81491803
3 -0.48624639
4 1.73167306
5 -1.31020021
6 -0.48514538
Coefficients of linear discriminants:
LD1 LD2 LD3 LD4 LD5
crim 0.26905214 -0.04038901 0.952783084 1.05716987 0.83157632
zn 0.02554270 -1.56332967 0.861975978 -1.07176445 0.45885648
indus 0.26308114 0.34083276 0.299613031 -0.64713459 -0.27149224
nox -0.03462213 0.07174078 -0.011080924 -0.44577196 0.17727824
rm -0.09400180 -0.18081151 -0.748312466 -0.28307867 0.35366014
age 0.09743790 0.59122392 -0.124953000 -0.52102866 0.63971252
dis -0.28508543 -0.26165892 -0.004961456 0.09291479 -0.31814880
rad 5.99463004 -1.10320047 -1.528777174 0.82447556 -1.05043782
tax 0.09961052 -0.43262170 0.468017911 -0.81595269 0.38580292
ptratio 0.25749486 0.08897531 0.282642121 -0.34518855 -0.02514549
black -0.04609198 -0.01324538 -0.083403025 0.01031609 -0.10644781
lstat 0.09758566 0.09732902 0.327335484 0.20747689 0.64677037
medv -0.06198719 -0.18784402 -0.206662871 0.28964586 0.98112363
Proportion of trace:
LD1 LD2 LD3 LD4 LD5
0.8529 0.0830 0.0298 0.0185 0.0159 
Super-Bonus
Data points are of course in same positions. Grouping differs slightly in main group if colours are coded either by crime or by cluster. In the separate group high-crime is well isolated whereas in clusters, there are two of them. If colours are coded by crime, particularly the high-crime is better gathered to one group.